Quantitative phase spectroscopy is presented as a novel method of measuring the wavelength-dependent refractive index of microscopic volumes. Light from a broadband source is filtered to an ~5 nm bandwidth and rapidly tuned across the visible spectrum in 1 nm increments by an acousto-optic tunable filter (AOTF). Quantitative phase images of semitransparent samples are recovered at each wavelength using off-axis interferometry and are processed to recover relative and absolute dispersion measurements. We demonstrate the utility of this approach by (i) spectrally averaging phase images to reduce coherent noise, (ii) measuring absorptive and dispersive features in microspheres, and (iii) quantifying bulk hemoglobin concentrations by absolute refractive index measurements. Considerations of using low coherence illumination and the extension of spectral techniques in quantitative phase measurements are discussed.